A uniform circular wire of radius \(R\) is forced to rotate about a fixed vertical diameter at constant angular velocity \(\omega\). A bead of mass \(m\) experiences gravity, is smoothly threaded on the wire and can slide without friction. Is the system holonomic? Construct a Lagrangian for the system. Show that there is just one solution for the off-axis motion of the bead where it does not oscillate. Does the system conserve energy?

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R- bead gravity